\(\theta\)-summability of Fourier series. (English) Zbl 1060.42021

In this paper the author investigates a general summability method of orthogonal series with the help of an integrable function \(\theta\). Under some conditions he shows that if the maximal Fejér operator is bounded from a Banach space \(\mathbf X\) to \(\mathbf Y\), then the maximal \(\theta\)-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh-Kaczmarz, Vilenkin and Ciesielski-Fourier series and Fourier transforms are considered. It is proved that the maximal operator of the \(\theta\)-means of these Fourier series is bounded from \(H_p\) to \(L_p\) for any \(1/2<p\leq\infty\) and is of weak type \((1,1)\). In the endpoint case \(p=1/2\) a weak type inequality is derived. As a consequence the author proves that the \(\theta\)-means of an integrable function converge a.e. to the function. Some special cases, such as the Weierstrass, Picard, Bessel, Riesz, de la Valée-Poussin, Rogosinski and Riemann summations are considered. Similar results are verified for several-dimensional Fourier series and Hardy spaces.


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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